Space vector modulation for matrix converter and current source converter

ABSTRACT

A converter includes a transformer including primary windings and secondary windings, switches connected to the primary windings, an output inductor connected to the secondary windings, and a controller connected to the switches. The controller turns the switches on and off based on dwell times calculated using space vector modulation with a reference current ref whose magnitude changes with time.

BACKGROUND OF THE INVENTION 1. Field of the Invention

The present invention relates to space vector modulation (SVM). Morespecifically, the present invention relates to an improved SVM algorithmthat can be used with matrix rectifiers, current-source rectifiers, andcurrent-source inverters.

2. Description of the Related Art

FIG. 1 shows an isolated matrix rectifier, FIG. 2 shows a current-sourcerectifier, and FIG. 3 shows a current-source inverter. Each of thecircuits shown in FIGS. 1-3 can be used either with known SVM methodsdiscussed in this section or with the novel SVM methods according to thepreferred embodiments of the present invention discussed in the DetailedDescription of Preferred Embodiments section below.

In FIG. 1, “line side” refers to the portion of the circuit on theleft-hand side of the transformer T_(r) that is connected to the linevoltages u_(a), u_(b), u_(c) for each of the phases A, B, C, and “loadside” refers to the portion of the circuit on the right-hand side of thetransformer T_(r) that is connected to the output voltage u_(o), i.e.,the load. On the line side, the three-phase AC current is combined intosingle-phase AC current, and on the load side, the single-phase ACcurrent is rectified by diodes D₁ to D₄ to provide DC current.

The isolated matrix rectifier includes a filter inductor L_(f) and afilter capacitor C_(f) that define a line-side filter that reduces thetotal harmonic distortion (THD), bi-directional switches S₁ to S₆arranged in a bridge as a 3-phase-to-1-phase matrix converter, atransformer T_(r) that provides high-voltage isolation between theline-side circuit and the load-side circuit, four diodes D₁ to D₄arranged in a bridge to provide output rectification, an output inductorL_(o), and an output capacitor C_(o) that define a filter for the outputvoltage. Bi-directional switches are used in this isolated matrixrectifier to open or close the current path in either direction. Asshown in FIG. 1, the bi-directional switch includes two uni-directionalswitches connected in parallel.

THD is defined as the ratio of the RMS amplitude of the higher harmonicfrequencies to the RMS amplitude of the fundamental frequency:

$\begin{matrix}{{THD} = \frac{\sqrt{\Sigma_{k = 2}^{\infty}V_{k}^{2}}}{V_{1}}} & (1)\end{matrix}$where V₁ is the amplitude of the fundamental frequency and V_(k) is theamplitude of the higher harmonic frequencies. It is desirable to reducethe THD because the harmonic current can be injected back into the powersystem.

SVM is an algorithm for the pulse-width modulation (PWM) of thebi-directional switches S₁ to S₆. That is, SVM is used to determine whenthe bi-directional switches S₁ to S₆ should be turned on and off. Thebi-directional switches S₁ to S₆ are controlled by digital signals,e.g., either one or zero. Typically, a one means the switch is on, and azero means the switch is off. In PWM, the width of the on signal, whichcontrols how long a switch is turned on, is modulated, or changed.

In known SVM, the main assumption is that the DC current is constant,which requires that the load-side inductor L_(o) should be infinite intheory and that the power converter should be only used incontinuous-conduction mode (CCM) operation. CCM occurs when the currentthrough the load-side inductor L₀ is always above zero. In contrast toCCM, discontinuous-conduction mode (DCM) occurs when the current throughthe load-side inductor L_(o) can be zero. The problem with using knownSVM with DCM is large THD, as shown in FIG. 10E. It is impossible toprovide an infinite load-side inductor L_(o) in practice. Although it ispossible to provide a load-side inductor L_(o) with a very largeinductance, doing so requires providing a large inductor that makesdesign difficult. It is impractical to assume that a power converterwill only be used in CCM operation in any application that includeslight-load conditions in which the power converter can be in DCMoperation.

For the isolated matrix rectifier shown in FIG. 1, a switching functionS_(i) can be defined as:

$\begin{matrix}{S_{i} = \left\{ {{\begin{matrix}{1,{S_{i}\mspace{14mu}{turn}\mspace{14mu}{on}}} \\{0,{S_{i}\mspace{14mu}{turn}\mspace{14mu}{off}}}\end{matrix}i} \in \left\{ {1,2,3,4,5,6} \right\}} \right.} & (2)\end{matrix}$

where S_(i) is the switching function for the i^(th) switch. Forexample, if S₁=1, then switch S₁ is on, and if S₁=0, then switch S₁ isoff.

Only two switches can be turned on at the same time to define a singlecurrent path. For example, if switches S₁ and S₆ are on, a singlecurrent path is defined between phases A and B through the transformerT_(r). If only two switches can conduct at the same time, with oneswitch in the top half of the bridge (S₁, S₃, S₅) and with the otherswitch in the bottom half of the bridge (S₂, S₄, S₆), then there arenine possible switching states as listed in Tables 1 and 2, includingsix active switching states and three zero switching states. In Table 1,line currents i_(a), i_(b), i_(c) are the currents in phases A, B, C,and the line-side current i_(p) is the current through the primarywinding of the transformer T_(r). In Table 2, the transformer turnsratio k is assumed to be 1 so that the inductor current i_(L) is equalto the line-side current i_(p).

TABLE 1 Space Vectors, Switching States, and Phase Currents SpaceSwitching States Vector S₁ S₂ S₃ S₄ S₅ S₆ i_(a) i_(b) i_(c) I₁ 1 0 0 0 01 i_(p) −i_(p) 0 I₂ 1 1 0 0 0 0 i_(p) 0 −i_(p) I₃ 0 1 1 0 0 0 0 i_(p)−i_(p) I₄ 0 0 1 1 0 0 −i_(p) i_(p) 0 I₅ 0 0 0 1 1 0 −i_(p) 0 i_(p) I₆ 00 0 0 1 1 0 −i_(p) i_(p) I₇ 1 0 0 1 0 0 0 0 0 I₈ 0 0 1 0 0 1 0 0 0 I₉ 01 0 0 1 0 0 0 0

TABLE 2 Space Vectors, Switching States, and Phase Currents SpaceSwitching States Vector S₁ S₂ S₃ S₄ S₅ S₆ i_(a) i_(b) i_(c) I₁ 1 0 0 0 01 i_(L) −i_(L) 0 I₂ 1 1 0 0 0 0 i_(L) 0 −i_(L) I₃ 0 1 1 0 0 0 0 i_(L)−i_(L) I₄ 0 0 1 1 0 0 −i_(L) i_(L) 0 I₅ 0 0 0 1 1 0 −i_(L) 0 i_(L) I₆ 00 0 0 1 1 0 −i_(L) i_(L) I₇ 1 0 0 1 0 0 0 0 0 I₈ 0 0 1 0 0 1 0 0 0 I₉ 01 0 0 1 0 0 0 0

The active and zero switching states can be represented by active andzero vectors. A vector diagram is shown in FIG. 4, with the six activevectors {right arrow over (I)}₁˜{right arrow over (I)}₆ and the threezero vectors {right arrow over (I)}₇˜{right arrow over (I)}₉. The activevectors {right arrow over (I)}₁˜{right arrow over (I)}₆ form a regularhexagon with six equal sectors I-VI, and the zero vectors {right arrowover (I)}₇˜{right arrow over (I)}₉ lie at the center of the hexagon.

The relationship between the vectors and the switching states can bederived as follows.

Because the three phases A, B, C are balanced:i _(a)(t)+i _(b)(t)+i _(c)(t)=0  (3)where i_(a)(t), i_(b)(t), and i_(c)(t) are the instantaneous currents inthe phases A, B, and C. Using equation (3), the three-phase currentsi_(a)(t), i_(b)(t), and i_(c)(t) can be transformed into two-phasecurrents in the α-β plane using the following transformation:

$\begin{matrix}{\begin{bmatrix}{i_{\alpha}(t)} \\{i_{\beta}(t)}\end{bmatrix} = {{\frac{2}{3}\begin{bmatrix}1 & {- \frac{1}{2}} & {- \frac{1}{2}} \\0 & {- \frac{\sqrt{3}}{2}} & {- \frac{\sqrt{3}}{2}}\end{bmatrix}}\begin{bmatrix}{i_{a}(t)} \\{i_{b}(t)} \\{i_{c}(t)}\end{bmatrix}}} & (4)\end{matrix}$where i_(α)(t), i_(β)(t) are the instantaneous currents in the phases α,β. A current vector I(t) can be expressed in the α-β plane as:{right arrow over (I)}(t)=i _(α)(t)+ji _(β)(t)  (5){right arrow over (I)}(t)=⅔[i _(a)(t)e ^(j0) +i _(b)(t)e ^(j2π/3) +i_(c)(t)e ^(j4π/3)]  (6)where j is the imaginary number and e^(jx)=cos x+j sin x. Then theactive vectors in FIG. 4 are provided by:

$\begin{matrix}{{\overset{\rightarrow}{I_{k}} = {{\frac{2}{\sqrt{3}}\frac{I_{d}}{k}e^{j{({{{({k - 1})}\frac{\pi}{3}} - \frac{\pi}{6}})}}\mspace{14mu}{for}\mspace{14mu} k} = 1}},2,\ldots,6} & (7)\end{matrix}$

The isolated matrix rectifier's controller determines a referencecurrent {right arrow over (I)}_(ref) and calculates the on and off timesof the switches S₁ to S₆ to approximate the reference current {rightarrow over (I)}_(ref) to produce the line-side current i_(a) and i_(b).The reference current {right arrow over (I)}_(ref) preferably issinusoidal with a fixed frequency and a fixed magnitude: {right arrowover (I)}_(ref)=I_(ref)e^(jθ). The fixed frequency is preferably thesame as the fixed frequency of each of the three-phase i_(a)(t),i_(b)(t), and i_(c)(t) to reduce harmful reflections. The controllerdetermines the magnitude of the reference current {right arrow over(I)}_(ref) to achieve a desired output voltage u_(o). That is, thecontroller can regulate the output voltage u_(o) by varying themagnitude of the reference current {right arrow over (I)}_(ref).

The reference current {right arrow over (I)}_(ref) moves through the α-βplane. The angle θ is defined as the angle between the α-axis and thereference current {right arrow over (I)}_(ref). Thus, as the angle θchanges, the reference current {right arrow over (I)}_(ref) sweepsthrough the different sectors.

The reference current {right arrow over (I)}_(ref) can be synthesized byusing combinations of the active and zero vectors. Synthesized meansthat the reference current {right arrow over (I)}_(ref) can berepresented as a combination of the active and zero vectors. The activeand zero vectors are stationary and do not move in the α-β plane asshown in FIG. 4. The vectors used to synthesize the reference current{right arrow over (I)}_(ref) change depending on which sector thereference current {right arrow over (I)}_(ref) is located. The activevectors are chosen by the active vectors defining the sector. The zerovector is chosen for each sector by determining which on switch the twoactive vectors have in common and choosing the zero vector that alsoincludes the same on switch. Using the zero vectors allows the magnitudeof the line-side current i_(p) to be adjusted.

For example, consider when the current reference {right arrow over(I)}_(ref) is in sector I. The active vectors {right arrow over (I)}₁and {right arrow over (I)}₂ define sector I. The switch S₁ is on forboth active vectors {right arrow over (I)}₁ and {right arrow over (I)}₂.The zero vector {right arrow over (I)}₇ also has the switch S₁ on. Thus,when the reference current {right arrow over (I)}_(ref) is located insector I, the active vectors {right arrow over (I)}₁ and {right arrowover (I)}₂ and zero vector {right arrow over (I)}₇ are used tosynthesize the reference current {right arrow over (I)}_(ref), whichprovides the following equation, with the right-hand side of theequation resulting from vector {right arrow over (I)}₇ being a zerovector with zero magnitude:

$\begin{matrix}{{\overset{\rightarrow}{I}}_{ref} = {{{\frac{T_{1}}{T_{s}}{\overset{\rightarrow}{I}}_{1}} + {\frac{T_{2}}{T_{s}}{\overset{\rightarrow}{I}}_{2}} + {\frac{T_{7}}{T_{s}}{\overset{\rightarrow}{I}}_{7}}} = {{\frac{T_{1}}{T_{s}}{\overset{\rightarrow}{I}}_{1}} + {\frac{T_{2}}{T_{s}}{\overset{\rightarrow}{I}}_{2}}}}} & (8)\end{matrix}$where T₁, T₂, and T₀ are the dwell times for the corresponding activeswitches and T_(s) is the sampling period.

The dwell time is the on time of the corresponding switches. Forexample, T₁ is the on time of the switches S₁ and S₆ for the activevector I₁. Because switch S₁ is on for each of vectors {right arrow over(I)}₁, {right arrow over (I)}₂, and {right arrow over (I)}₇, the switchS₁ is on the entire sampling period T_(s). The ratio T₁/T_(s) is theduty cycle for the switch S₆ during the sampling period T_(s).

The sampling period T_(s) is typically chosen such that the referencecurrent {right arrow over (I)}_(ref) is synthesized multiple times persector. For example, the reference current {right arrow over (I)}_(ref)can be synthesized twice per sector so that the reference current {rightarrow over (I)}_(ref) is synthesized twelve times per cycle, where onecomplete cycle is when the reference current {right arrow over(I)}_(ref) goes through sectors I-VI.

The dwell times can be calculated using the ampere-second balancingprinciple, i.e., the product of the reference current {right arrow over(I)}_(ref) and sampling period T_(s) equals the sum of the currentvectors multiplied by the time interval of synthesizing space vectors.Assuming that the sampling period T_(s) is sufficiently small, thereference current {right arrow over (I)}_(ref) can be consideredconstant during sampling period T_(s). The reference current {rightarrow over (I)}_(ref) can be synthesized by two adjacent active vectorsand a zero vector. For example, when the reference current {right arrowover (I)}_(ref) is in sector I as shown in FIG. 5, the reference current{right arrow over (I)}_(ref) can be synthesized by vectors {right arrowover (I)}₁, {right arrow over (I)}₂, and {right arrow over (I)}₇. Theampere-second balancing equation is thus given by the followingequations:{right arrow over (I)} _(ref) T _(s) ={right arrow over (I)} ₁ T ₁+{right arrow over (I)} ₂ T ₂ +{right arrow over (I)} ₇ T ₇  (9)T _(s) =T ₁ +T ₂ +T ₇  (10)where T₁, T₂, and T₇ are the dwell times for the vectors {right arrowover (I)}₁, {right arrow over (I)}₂, and {right arrow over (I)}₇ andT_(s) is sampling time. Then the dwell times are given by:

$\begin{matrix}{T_{1} = {{mT}_{s}\mspace{14mu}{\sin\left( {{\pi\text{/}6} - \theta} \right)}}} & (11) \\{T_{2} = {{mT}_{s}\mspace{14mu}{\sin\left( {{\pi\text{/}6} + \theta} \right)}}} & (12) \\{{T_{7} = {T_{s} - T_{1} - T_{2}}}{where}} & (13) \\{{m = {k\frac{I_{ref}}{i_{L}}}},} & (14)\end{matrix}$θ is sector angle between current reference {right arrow over(I)}_(ref), and α-axis shown in FIG. 5, and k is the transformer turnsratio.

However, the above dwell time calculations are based on the assumptionthat the inductor current i_(L) is constant. If the inductor currenti_(L) has ripples, the dwell time calculation based on these equationsis not accurate. The larger the ripple, the larger the error will be. Asa result, the THD of the line-side current will be increased. In actualapplications, the load-side inductance is not infinite, and the currentripple always exists. As shown in FIG. 6A, if the load-side inductanceis small, then the current ripple is too large to use known SVM. Asshown in FIG. 6B, to provide acceptable waveforms and to reduceline-side THD, the load-side inductance must be very large to reduce thecurrent ripple and to come as close as possible to a theoretical value.

Known SVM can also be applied to the current-source rectifier in FIG. 2and to the current-source inverter in FIG. 3 using the same techniquesas discussed above with respect to equations (9)-(14).

A large load-side inductance has the problems of large size, excessiveweight, and high loss, for example. The current ripple in a practicalinductor also has the problems in modulation signals using traditionalSVM, including increased line-side THD. In addition, DCM is unavoidablewhen the load varies. Under light loads, the load-side inductor L_(o)might be in DCM without a dummy load.

In known SVM for matrix rectifiers, current-source rectifiers, andcurrent-source inverters, the DC current is assumed to be constant orthe current ripple is assumed to be very small. Thus, known SVM includesat least the following problems:

-   -   1) The Load-side inductance must be large to maintain small        current ripple.    -   2) As a result of 1) the Load-side inductor size must be large.    -   3) Current ripple increases the THD of the line-side current.    -   4) Line-side current THD is high at light load.    -   5) Known SVM can only be used in CCM operation.

SUMMARY OF THE INVENTION

To overcome the problems described above, preferred embodiments of thepresent invention provide an improved SVM with the following benefits:

-   -   1) Reduced load-side inductance.    -   2) Reduced load-side inductor size.    -   3) Decreased THD of the line-side current, even with large        current ripple or light-load condition.    -   4) Improved SVM is capable of being used with both DCM and CCM        modes.    -   5) Improved SVM is simple and is capable of being calculated in        real time.

A preferred embodiment of the present invention provides a converterthat includes a transformer including primary windings and secondarywindings, switches connected to the primary windings, an output inductorconnected to the secondary windings, and a controller connected to theswitches. The controller turns the switches on and off based on dwelltimes calculated using space vector modulation with a reference current{right arrow over (I)}_(ref) whose magnitude changes with time.

Another preferred embodiment of the present invention provides acorresponding space-vector-modulation method.

Preferably, the switches include six switches; the space vectormodulation includes using six active switching states and three zeroswitching states; a current space is divided into six sectors by the sixactive switching states such that a vector with θ=0 is located halfwaybetween two of the active switching states; and magnitudes of the sixactive switching states change with time.

The controller preferably turns the six switches on and off based ondwell times that are calculated based on an ampere-second balanceequation:{right arrow over (I)}I _(ref) T _(s)=∫₀ ^(T) ^(α) {right arrow over(I)} _(α) dt+∫ ₀ ^(T) ^(β) {right arrow over (I)} _(β) dt+∫ ₀ ^(T) ^(o){right arrow over (I)} ₀ dtwhere {right arrow over (I)}_(ref)=I_(ref) e^(jθ), θ is an angle betweenthe reference current {right arrow over (I)}_(ref) and the vector withθ=0, T_(s) is a sampling period, {right arrow over (I)}_(α), {rightarrow over (I)}_(β), {right arrow over (I)}₀, are three nearest adjacentactive vectors to {right arrow over (I)}_(ref), and T_(α), T_(β), T₀ aredwell times of {right arrow over (I)}_(α), {right arrow over (I)}_(β),{right arrow over (I)}₀. The controller preferably turns the sixswitches on and off based on a vector sequence {right arrow over(I)}_(α), {right arrow over (I)}₀, −{right arrow over (I)}_(β), {rightarrow over (I)}₀, {right arrow over (I)}_(β), {right arrow over (I)}₀,−{right arrow over (I)}_(α), {right arrow over (I)}₀, during thesampling period T_(s). The controller preferably turns the six switcheson and off based on a timing sequence T_(α)/2, T₀/4, T_(β)/2, T₀/4,T_(β)/2, T₀/4, T_(α)/2, T₀/4, during the sampling period T_(s).

Preferably, the controller calculates the dwell times using:

$T_{\alpha} = \frac{{- B} + \sqrt{B^{2} + {{AC}\mspace{14mu}{\sin\left( {{\pi\text{/}6} - \theta} \right)}}}}{A}$$T_{\beta} = {\frac{{- B} + \sqrt{B^{2} + {{AC}\mspace{14mu}{\sin\left( {{\pi\text{/}6} - \theta} \right)}}}}{A} \cdot \frac{\sin\left( {{\pi\text{/}6} + \theta} \right)}{\sin\left( {{\pi\text{/}6} - \theta} \right)}}$T₀ = T_(s) − T_(α) − T_(β) where$A = {\left( {{4u_{1\alpha}\text{/}k} - u_{o}} \right) + {\left( {{4u_{1\beta}\text{/}k} - u_{o}} \right)\frac{\sin\left( {{\pi\text{/}6} + \theta} \right)}{\sin\left( {{\pi\text{/}6} - \theta} \right)}}}$B = 4L_(o)I_(L 0) − 3u_(o)T_(s)/2 C = 8kL_(o)I_(ref)T_(s)u_(1α) is a line-to-line voltage depending on the active switching state{right arrow over (I)}_(α), u_(1β) is a line-to-line voltage dependingon the active switching state {right arrow over (I)}_(β), k is atransformer turns ratio, u_(o) is an output voltage of the converter, θis the angle between the reference current {right arrow over (I)}_(ref)and the vector with θ=0, L_(o) is an inductance of the output inductor,I_(L0) is the current through inductor L_(o) at a beginning of thesampling period T_(s), T_(s) is the sampling period, and I_(ref) is amagnitude of the vector {right arrow over (I)}_(ref).

The controller preferably calculates the dwell times using:

$T_{\alpha} = {2\sqrt{\frac{{kI}_{ref}L_{o}T_{s}{\sin\left( {{\pi\text{/}6} - \theta} \right)}}{{u_{1\;\alpha}\text{/}k} - u_{o}}}}$$T_{\beta} = {2\sqrt{\frac{{kL}_{o}I_{ref}T_{s}{\sin\left( {{\pi\text{/}6} + \theta} \right)}}{{u_{1\;\beta}\text{/}k} - \mu_{o}}}}$where k is a transformer turns ratio, L_(o) is an inductance of theoutput inductor, I_(ref) is the magnitude of the vector {right arrowover (I)}_(ref), T_(s) is the sampling period, θ is an angle between thereference current {right arrow over (I)}_(ref) and the vector with θ=0,u_(1α) is a line-to-line voltage depending on the active switching state{right arrow over (I)}_(α), u_(1β) is a line-to-line voltage dependingon the active switching state {right arrow over (I)}_(β), and u_(o) isan output voltage of the converter.

The converter is preferably one of a matrix rectifier, a current-sourcerectifier, and a current-source inverter. The converter is preferablyoperated in a continuous-conduction mode or a discontinuous-conductionmode.

The above and other features, elements, characteristics, steps, andadvantages of the present invention will become more apparent from thefollowing detailed description of preferred embodiments of the presentinvention with reference to the attached drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a circuit diagram of an isolated matrix rectifier.

FIG. 2 is a circuit diagram of a current-source rectifier.

FIG. 3 is a circuit diagram of a current-source inverter.

FIG. 4 shows a current-space vector hexagon.

FIG. 5 shoes the synthesis of reference current {right arrow over(I)}_(ref) using I₁ and I₂ using known SVM.

FIG. 6 shows ideal and real DC current waveforms.

FIG. 7 shows the synthesis of reference current {right arrow over(I)}_(ref) using I_(α) and I_(β) using SVM of a preferred embodiment ofthe present invention.

FIG. 8 shows the waveforms of the isolated matrix rectifier shown inFIG. 1.

FIGS. 9A, 9C, and 9E show waveforms of the isolated matrix rectifiershown in FIG. 1 in CCM using known SVM, and FIGS. 9B, 9D, and 9F showcorresponding waveforms of the isolated matrix rectifier shown in FIG. 1in CCM using SVM according to various preferred embodiments of thepresent invention.

FIGS. 10A, 10C, and 10E show waveforms of the isolated matrix rectifiershown in FIG. 1 in DCM using known SVM, and FIGS. 10B, 10D, and 10F showcorresponding waveforms of the isolated matrix rectifier shown in FIG. 1in DCM using SVM according to various preferred embodiments of thepresent invention.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

Preferred embodiments of the present invention improve the known SVM.The improved SVM is capable of being used with both DCM and CCMoperation, is capable of being used with smaller load-side inductors,and reduces line-side THD.

As with the known SVM, the improved SVM includes nine switching states,including six active switching states and three zero switching states asshown in FIG. 4, that are used to synthesize the reference current{right arrow over (I)}_(ref) as shown in FIG. 7. However, in theimproved SVM, the six active switching states, although stationary, areassumed to change with time. That is, the magnitude of the activeswitching states changes with time which is true in actual application.

The reference current {right arrow over (I)}_(ref) preferably issynthesized by the three nearest vectors {right arrow over (I)}_(α),{right arrow over (I)}_(β), {right arrow over (I)}₀ as shown in FIG. 7,and the dwell time of each vector is T_(α), T_(β), T₀. Here, (α, β)represent the subscript of the pair of active vectors in each sectorsuch as (1,2) or (2,3) or (3,4) or (5,6) or (6,1). The dwell timespreferably are calculated based on the principle of ampere-secondbalance. Because of current ripple, the inductor current is notconstant, so the ampere-second balance of equation (9) becomes:{right arrow over (I)} _(ref) T _(s)=∫₀ ^(T) ^(α) {right arrow over (I)}_(α) dt+∫ ₀ ^(T) ^(β) {right arrow over (I)} _(β) dt+∫ ₀ ^(T) ⁰ {rightarrow over (I)} ₀ dt  (15)

Applying equation (15) to the isolated matrix rectifier shown in FIG. 1,provides the following analysis. The following assumptions are made inthe following analysis:

-   -   1) Transformer T_(r) is ideal; and    -   2) In one sampling period T_(s), phase voltages u_(a), u_(b),        u_(c) are constant.

Because of the isolation provided by the transformer, the output voltageof the matrix converter u₁(t) must alternate between positive andnegative with high frequency to maintain volt-sec balance. Thus, thepreferred vector sequence in every sampling period T_(s) is divided intoeight segments as İ_(α), İ₀, −{right arrow over (I)}_(β), İ₀, {rightarrow over (I)}_(β), İ₀, −İ_(α), İ₀, and the dwell time of each vectoris respectively T_(α)/2, T₀/4, T_(β)/2, T₀/4, T_(β)/2, T₀/4, T_(α)/2,T₀/4. However, the sequence of the active vectors and zero vectors canbe combined in different ways, and the dwell time for the zero vectorsis not necessary to be divided equally. For example, the vector sequencecould be six segments as {right arrow over (I)}_(α), {right arrow over(I)}_(β), {right arrow over (I)}₀, −{right arrow over (I)}_(β), {rightarrow over (I)}₀, with dwell time T_(α)/2, T_(β)/2, T₀/2, T_(α)/2,T_(β)/2, T₀/2, respectively. Only the case with eight segments as {rightarrow over (I)}_(α), {right arrow over (I)}₀, −{right arrow over(I)}_(β), {right arrow over (I)}₀, {right arrow over (I)}_(β), {rightarrow over (I)}₀, −{right arrow over (I)}_(α), {right arrow over (I)}₀,and the dwell time of each vector with T_(α)/2, T₀/4, T_(β)/2, T₀/4,T_(β)/2, T₀/4, T_(α)/2, T₀/4 is used as an example to show how the dwelltimes can be calculated to eliminate the effect of the current ripple onload side. FIG. 8 shows the waveforms of the matrix converter outputvoltage u₁(t) the inductor current i_(L)(t), the matrix converter outputcurrent i_(p)(t), and the phase current i_(a)(t). The inductor currenti_(L)(t) at the time t₀, t₁, t₂, t₃, t₄, t₅, t₆, t₇, t₈, where t₁ and(t₇−t₆)=T_(α)/2, (t₃−t₂) and (t₅−t₄)=T_(β)/2, and the dwell time of thezero vectors are all T₀/4, can be described in equation (16):

$\begin{matrix}{{I_{Li} = {{I_{{Li} - 1} + {{\frac{u_{Li}}{L_{o}} \cdot \left( {t_{i} - t_{i - 1}} \right)}\mspace{14mu} i}} = 1}},2,3,4,5,6,7,8} & (16)\end{matrix}$where the u_(Li) is the voltage of load-side inductor between timest_(i-1) and t_(i) and L_(o) is the inductance of the load-side inductorL_(o). The instantaneous value of the load-side inductor current isprovided by:

$\begin{matrix}{{i_{L}(t)} = {{I_{{Li} - 1} + {{\frac{u_{Li}}{L_{o}} \cdot \left( {t - t_{i - 1}} \right)}\mspace{14mu} t_{i - 1}}} < t < t_{i}}} & (17)\end{matrix}$

The output current i_(p) of the matrix converter is provided by:

$\begin{matrix}{{i_{p}(t)} = \left\{ {\begin{matrix}{{{i_{L}(t)}\text{/}k}\mspace{14mu}} & {u_{1} > 0} \\{{- {i_{L}(t)}}\text{/}k} & {u_{1} < 0}\end{matrix} = {{{gi}_{L}(t)}\text{/}k}} \right.} & (18)\end{matrix}$where k is turns ratio of the transformer and the sign function g isdefined by:

$\begin{matrix}{g = \left\{ {\begin{matrix}{1\mspace{14mu}} & {u_{1} > 0} \\{- 1} & {u_{1} < 0}\end{matrix},} \right.} & (19)\end{matrix}$Using equation (18), equation (7) for the active vectors becomes:{right arrow over (I)} _(k)2/√{square root over (3)}i _(p)(t)e^(j((k-1)π/3-π/6)) k−1,2,3,4,5,6  (20)Substituting equations (17), (18), and (20) into the ampere-secondbalancing equation (15) provides:

$\begin{matrix}\begin{matrix}{{{\overset{\rightharpoonup}{I}}_{ref}T_{s}} = {{\int_{0}^{t_{1}}{{\overset{\rightharpoonup}{I}}_{\alpha +}{dt}}} + {\int_{t_{2}}^{t_{3}}{{\overset{\rightharpoonup}{I}}_{\beta -}\ {dt}}} + {\int_{t_{4}}^{t_{5}}{{\overset{\rightharpoonup}{I}}_{\beta +}{dt}}} + {\int_{t_{6}}^{t_{7}}{{\overset{\rightharpoonup}{I}}_{\alpha -}{dt}}}}} \\{= {{\int_{0}^{T_{\alpha}\text{/}2}{{\overset{\rightharpoonup}{I}}_{\alpha +}{dt}}} + {\int_{0}^{T_{\beta}\text{/}2}{{\overset{\rightharpoonup}{I}}_{\beta -}{dt}}} + {\int_{0}^{T_{\beta}\text{/}2}{{\overset{\rightharpoonup}{I}}_{\beta +}{dt}}} + {\int_{0}^{T_{\alpha}\text{/}2}{{\overset{\rightharpoonup}{I}}_{\alpha -}{dt}}}}} \\{= {{\int_{0}^{T_{\alpha}\text{/}2}{{\frac{2}{\sqrt{3}} \cdot \frac{1}{k}}\left( {I_{L\; 0} + {\frac{u_{L\; 1}}{L_{o}}t}} \right)e^{j{({{{\alpha\pi}\text{/}3} - {\pi\text{/}6}})}}{dt}}} + {\int_{0}^{T_{\beta}\text{/}2}{{\frac{2}{\sqrt{3}} \cdot \frac{1}{k}}\left( {I_{L\; 2} + {\frac{u_{L\; 3}}{L_{o}}t}} \right)e^{j{({{{\beta\pi}\text{/}3} - {\pi\text{/}6}})}}{dt}}} +}} \\{{\int_{0}^{T_{\beta}\text{/}2}{{\frac{2}{\sqrt{3}} \cdot \frac{1}{k}}\left( {I_{L\; 4} + {\frac{u_{L\; 5}}{L_{o}}t}} \right)e^{j{({{{\beta\pi}\text{/}3} - {\pi\text{/}6}})}}{dt}}} + {\int_{0}^{T_{\alpha}\text{/}2}{{\frac{2}{\sqrt{3}} \cdot \frac{1}{k}}\left( {I_{L\; 6} + {\frac{u_{L\; 7}}{L_{o}}t}} \right)e^{j{({{{\beta\pi}\text{/}3} - {\pi\text{/}6}})}}{dt}}}} \\{= {{\int_{0}^{T_{\alpha}\text{/}2}{{\frac{2}{\sqrt{3}} \cdot \frac{1}{k}}\left( {I_{L\; 0} + i_{L\; 6} + {\frac{2u_{L\; 1}}{L_{o}}t}} \right)e^{j{({{{\alpha\pi}\text{/}3} - {\pi\text{/}6}})}}{dt}}} + {\int_{0}^{T_{\beta}\text{/}2}{{\frac{2}{\sqrt{3}} \cdot \frac{1}{k}}\left( {I_{L\; 2} + i_{L\; 4} + {\frac{2u_{L\; 2}}{L_{o}}t}} \right)e^{j{({{{\beta\pi}\text{/}3} - {\pi\text{/}6}})}}{dt}}}}} \\{= {{{\frac{2}{\sqrt{3}} \cdot \frac{1}{k}}\left( {{\left( {I_{L\; 0} + I_{L\; 6}} \right)\frac{T_{\alpha}}{2}} + {\frac{u_{L\; 1}}{L_{o}}\left( \frac{T_{\alpha}}{2} \right)^{2}}} \right)e^{j{({{{\alpha\pi}\text{/}3} - {\pi\text{/}6}})}}} + {{\frac{2}{\sqrt{3}} \cdot \frac{1}{k}}\left( {{\left( {I_{L\; 2} + I_{L\; 4}} \right)\frac{T_{\beta}}{2}} + {\frac{u_{L\; 2}}{L_{o}}\left( \frac{T_{\beta}}{2} \right)^{2}}} \right)e^{j{({{{\beta\pi}\text{/}3} - {\pi\text{/}6}})}}}}}\end{matrix} & (21)\end{matrix}$where (α, β) can be (1,2) or (2,3) or (3,4) or (5,6) or (6,1), dependingon which sector {right arrow over (I)}_(ref) is located in. For example,if {right arrow over (I)}_(ref) is located in sector I, (α, β) will be(1,2).

Substituting {right arrow over (I)}_(ref)=I_(ref)e^(jθ) into equation(21), the dwell times can be calculated under the following threedifferent cases.

Case 1: when the inductance L_(o)>∞ or the inductance L_(o) is so largethat the current ripple can be ignored so thati_(L0)=i_(L2)=i_(L4)=i_(L6)−I_(L), then the dwell times are the same asthe known SVM.T _(α) =mT _(s) sin(π/6−θ)  (22)T _(β) =mT _(s) sin(π/6+θ)  (23)T ₀ =T _(s) −T _(α) −T _(β)  (24)where the modulation index m is given by:

$\begin{matrix}{{m = {k\frac{I_{ref}}{I_{L}}}},} & (25)\end{matrix}$and θ is the angle between the reference current {right arrow over(I)}_(ref) and the α-axis as shown in FIG. 7.

In this case, the improved SVM according to various preferredembodiments of the present invention is consistent with the known SVM.

Case 2: When the inductance L_(o) is very small or the load is verylight, then the load-side can be in DCM mode. The dwell times arecalculated as:

$\begin{matrix}{T_{\alpha} = {2\sqrt{\frac{{kI}_{ref}L_{o}T_{s}{\sin\left( {{\pi\text{/}6} - \theta} \right)}}{{u_{1\;\alpha}\text{/}k} - u_{o}}}}} & (26) \\{T_{\beta} = {2\sqrt{\frac{{kL}_{o}I_{ref}T_{s}{\sin\left( {{\pi\text{/}6} + \theta} \right)}}{{\beta_{1\;\beta}\text{/}k} - u_{o}}}}} & (27) \\{T_{0} = {T_{s} - T_{\alpha} - T_{\beta}}} & (28)\end{matrix}$where k is the transformer turns ratio, L_(o) is the inductance of theload-side inductor L₀, {right arrow over (I)}_(ref) is the magnitude ofthe vector {right arrow over (I)}_(ref) and is determined by thecontroller, T_(s) is the sampling period, θ is the angle between thereference current {right arrow over (I)}_(ref) and the a-axis as shownin FIG. 7, u_(1α) is measured by the controller and corresponds to aline-to-line voltage depending on the switching state, u_(1β) ismeasured by the controller and corresponds to a line-to-line voltagedepending on the switching state, and u_(o) is the output voltage asmeasured by the controller. The line-to-line voltages u_(1α) and u_(1β)depend on the switching state. For example, in Sector I with activevectors {right arrow over (I)}₁ and {right arrow over (I)}₂,line-to-line voltages u_(1α) and u_(1β) are u_(ab) and u_(ac),respectively.

Case 3: when in CCM operation and the current ripple cannot be ignored,then the dwell times are calculated as:

$\begin{matrix}{T_{\alpha} = \frac{{- B} + \sqrt{B^{2} + {{AC}\mspace{14mu}{\sin\left( {{\pi\text{/}6} - \theta} \right)}}}}{A}} & (29) \\{T_{\beta} = {\frac{{- B} + \sqrt{B^{2} + {{AC}\mspace{14mu}{\sin\left( {{\pi\text{/}6} - \theta} \right)}}}}{A} \cdot \frac{\sin\left( {{\pi\text{/}6} + \theta} \right)}{\sin\left( {{\pi\text{/}6} - \theta} \right)}}} & (30) \\{{T_{0} = {T_{s} - T_{\alpha} - T_{\beta}}}{where}} & (31) \\{A = {\left( {{4u_{1\alpha}\text{/}k} - u_{o}} \right) + {\left( {{4u_{1\beta}\text{/}k} - u_{o}} \right)\frac{\sin\left( {{\pi\text{/}6} + \theta} \right)}{\sin\left( {{\pi\text{/}6} - \theta} \right)}}}} & (32) \\{B = {{2{nL}_{o}I_{L\; 0}} - {3u_{o}T_{s}\text{/}2}}} & (33) \\{C = {2n^{2}{kLI}_{ref}T_{s}}} & (34)\end{matrix}$where u_(1α) is measured by the controller and corresponds to aline-to-line voltage depending on the switching state, u_(1β) ismeasured by the controller and corresponds to a line-to-line voltagedepending on the switching state, k is the transformer turns ratio,u_(o) is the output voltage as measured by the controller, θ is theangle between the reference current {right arrow over (I)}_(ref) and theα-axis as shown in FIG. 7, L_(o) is the inductance of the load-sideinductor L_(o), I_(L0) is the current through inductor L_(o) as measuredby the controller at the beginning of the sampling period T_(s), T_(s)is the sampling period, and I_(ref) is the magnitude of the vector{right arrow over (I)}_(ref) and is determined by the controller. In onesampling period T_(s), the vector I_(α) is divided to n equal parts. Inthis example, n is 2 because one sampling period includes I_(α) and−I_(α). If n=2, then B and C are provided by:B=4L _(o) I _(L0)3u _(o) T _(s)/2  (35)C=8kL _(o) I _(ref) T _(s)  (36)

FIGS. 9A, 9C, and 9E show waveforms of the isolated matrix rectifiershown in FIG. 1 in CCM using known SVM, and FIGS. 9B, 9D, and 9F showcorresponding waveforms of the isolated matrix rectifier shown in FIG. 1in CCM using SVM according to various preferred embodiments of thepresent invention. In FIGS. 9A and 9B, the load-side inductor current iscontinuous, so the isolated matrix rectifier is operating in CCM. FIGS.9C and 9D show the waveforms in the time domain, and FIGS. 9E and 9Fshow the waveforms in the frequency domain. Comparing these figuresdemonstrates that the improved SVM according to various preferredembodiments of the present invention provide a line-side current with abetter shaped waveform and with a smaller THD. The THD using theimproved SVM was measured as 4.71% while the THD using the known SVM wasmeasured as 7.59%, for example.

FIGS. 10A, 10C, and 10E show waveforms of the isolated matrix rectifiershown in FIG. 1 in DCM using known SVM, and FIGS. 10B, 10D, and 10F showcorresponding waveforms of the isolated matrix rectifier shown in FIG. 1in DCM using SVM according to various preferred embodiments of thepresent invention. In FIGS. 10A and 10, the load-side inductor currentis discontinuous (i.e., the current is equal to zero), so the isolatedmatrix rectifier is operating in DCM. FIGS. 10C and 10D show thewaveforms in time domain, and FIGS. 10E and 10F show the waveforms inthe frequency domain. Comparing these figures demonstrates that theimproved SVM according to various preferred embodiments of the presentinvention provide a line-side current with a better shaped waveform andwith a smaller THD. The THD using the improved SVM was measured as 6.81%while the THD using known SVM was measured as 17.4%, for example.

Thus, the improved SVM according to various preferred embodiments of thepresent invention is capable of being used with the isolated matrixrectifier in FIG. 1 in both CCM and DCM operation. The line-side currentTHD is significantly reduced with the improved SVM compared to knownSVM. The improved SVM is suitable for the compact and high-efficiencydesign with a wide-load range. The improved SVM can also be applied tocurrent-source converter to improve the AC side current THD.

In the preferred embodiments of the present, to calculate the dwelltimes, the controller measures transformer primary current i_(p) (orinductor current I_(L)), line voltages u_(a), u_(b), u_(c), and outputvoltage u_(o). The controller can be any suitable controller, including,for example, a PI controller, a PID controller, etc. The controller canbe implemented in an IC device or a microprocessor that is programmed toprovide the functions discussed above.

The same techniques and principles applied to the isolated matrixrectifier in FIG. 1 can also be applied to the current-source rectifierin FIG. 2 and to the current-source inverter in FIG. 3. These techniquesand principles are not limited to the devices shown in FIGS. 1-3 and canbe applied to other suitable devices, including, for example,non-isolated devices.

It should be understood that the foregoing description is onlyillustrative of the present invention. Various alternatives andmodifications can be devised by those skilled in the art withoutdeparting from the present invention. Accordingly, the present inventionis intended to embrace all such alternatives, modifications, andvariances that fall within the scope of the appended claims.

What is claimed is:
 1. A converter comprising: a transformer includingprimary windings and secondary windings; switches connected to theprimary windings; an output inductor connected to the secondarywindings; and a controller connected to the switches; wherein thecontroller turns the switches on and off based on dwell times calculatedusing space vector modulation with a reference current

_(ref) whose magnitude changes with time.
 2. A converter of claim 1,wherein: the switches include six switches; the space vector modulationincludes using six active switching states and three zero switchingstates; a current space is divided into six sectors by the six activeswitching states such that a vector with θ=0 is located halfway betweentwo of the active switching states; and magnitudes of the six activeswitching states change with time.
 3. A converter of claim 2, wherein:the controller turns the six switches on and off based on dwell timesthat are calculated based on an ampere-second balance equation:

_(ref) T _(s)=∫₀ ^(T) ^(α)

_(α) dt+∫ ₀ ^(T) ^(β)

_(β) dt+∫ ₀ ^(T) ⁰

₀ dt where {right arrow over (I)}_(ref)=I_(ref) e^(jθ), θ is an anglebetween the reference current

_(ref) and the vector with θ=0, T_(s) is a sampling period,

_(α),

_(β),

₀, are three nearest adjacent active vectors to

_(ref), and T_(α), T_(β), T₀ are dwell times of

_(α),

_(β),

₀.
 4. A converter of claim 3, wherein the controller turns the sixswitches on and off based on a vector sequence

_(α),

₀, −

_(β),

₀,

_(β),

₀, −

_(α),

₀, during the sampling period T_(s).
 5. A converter of claim 4, whereinthe controller turns the six switches on and off based on a timingsequence T_(α)/2, T₀/4, T_(β)/2, T₀/4, T_(β)/2, T₀/4, T_(α)/2, T₀/4,during the sampling period T_(s).
 6. A converter of claim 4, wherein:the controller calculates the dwell times using:$T_{\alpha} = \frac{{- B} + \sqrt{B^{2} + {{AC}\mspace{14mu}{\sin\left( {{\pi\text{/}6} - \theta} \right)}}}}{A}$$T_{\beta} = {\frac{{- B} + \sqrt{B^{2} + {{AC}\mspace{14mu}{\sin\left( {{\pi\text{/}6} - \theta} \right)}}}}{A} \cdot \frac{\sin\left( {{\pi\text{/}6} + \theta} \right)}{\sin\left( {{\pi\text{/}6} - \theta} \right)}}$T₀ = T_(s) − T_(α) − T_(β) where$A = {\left( {{4u_{1\alpha}\text{/}k} - u_{o}} \right) + {\left( {{4u_{1\beta}\text{/}k} - u_{o}} \right)\frac{\sin\left( {{\pi\text{/}6} + \theta} \right)}{\sin\left( {{\pi\text{/}6} - \theta} \right)}}}$B = 4L_(o)I_(L 0) − 3u_(o)T_(s)/2 C = 8kL_(o)I_(ref)T_(s) u_(1α) is aline-to-line voltage depending on the active switching state {rightarrow over (I)}_(a), u_(1β) is a line-to-line voltage depending on theactive switching state Î_(β), k is a transformer turns ratio, u_(o) isan output voltage of the converter, θ is the angle between the referencecurrent

_(ref) and the vector with θ=0, L_(o) is an inductance of the outputinductor, I_(L0) is the current through inductor L_(o) at a beginning ofthe sampling period T_(s), T_(s) is the sampling period, and I_(ref) isa magnitude of the vector {right arrow over (I)}_(ref).
 7. A converterof claim 4, wherein: the controller calculates the dwell times using:$T_{\alpha} = {2\sqrt{\frac{{kI}_{ref}L_{o}T_{s}{\sin\left( {{\pi\text{/}6} - \theta} \right)}}{{u_{1\;\alpha}\text{/}k} - u_{o}}}}$$T_{\beta} = {2\sqrt{\frac{{kL}_{o}I_{ref}T_{s}{\sin\left( {{\pi\text{/}6} - \theta} \right)}}{{u_{1\;\beta}\text{/}k} - u_{o}}}}$where k is a transformer turns ratio, L_(o) is an inductance of theoutput inductor, I_(ref) is the magnitude of the vector {right arrowover (I)}_(ref), T_(s) is the sampling period, θ is an angle between thereference current

_(ref) and the vector with θ=0, u_(1α) is a line-to-line voltagedepending on the active switching state {right arrow over (I)}_(α),u_(1β) is a line-to-line voltage depending on the active switching state{right arrow over (I)}_(β), and u_(o) is an output voltage of theconverter.
 8. A converter of claim 1, wherein the converter is one of amatrix rectifier, a current-source rectifier, and a current-sourceinverter.
 9. A converter of claim 1, wherein the converter is operatedin a continuous-conduction mode.
 10. A converter of claim 1, wherein theconverter is operated in a discontinuous-conduction mode.
 11. Aspace-vector-modulation method for a converter including a transformerwith primary windings and secondary windings, switches connected to theprimary windings, and an output inductor connected to the secondarywindings, the space-vector-modulation method comprising: turning theswitches on and off based on dwell times calculated using space vectormodulation with a reference current {right arrow over (I)}_(ref) whosemagnitude changes with time.
 12. A method of claim 11, wherein: theswitches include six switches; calculating the dwell times uses: sixactive switching states and three zero switching states; and a currentspace that is divided into six sectors by the six active switchingstates such that a vector with θ=0 is located halfway between two of theactive switching states; and magnitudes of the six active switchingstates change with time.
 13. A method of claim 12, wherein: turning thesix switches on and off is based on dwell times that are calculatedbased on an ampere-second balance equation:

_(ref) T _(s)=∫₀ ^(T) ^(α)

_(α) dt+∫ ₀ ^(T) ^(β)

_(β) dt+∫ ₀ ^(T) ⁰

₀ dt where {right arrow over (I)}_(ref)=I_(ref) e^(jθ), θ is an anglebetween the reference current

_(ref) and the vector with θ=0, T_(s) is a sampling period,

_(α),

_(β),

₀, are three nearest adjacent active vectors to

_(ref), and T_(α), T_(β), T₀ are dwell times of

_(α),

_(β),

₀.
 14. A method of claim 13, wherein turning the six switches on and offis based on a vector sequence

_(α),

₀, −

_(β),

₀,

_(β),

₀, −

_(α),

₀, during the sampling period T_(s).
 15. A method of claim 14, whereinturning the six switches on and off is based on a timing sequenceT_(α)/2, T₀/4, T_(β)/2, T₀/4, T_(β)/2, T₀/4, T_(α)/2, T₀/4, during theperiod T_(s).
 16. A method of claim 14, wherein: the dwell times arecalculated using:$T_{\alpha} = \frac{{- B} + \sqrt{B^{2} + {{AC}\mspace{14mu}{\sin\left( {{\pi\text{/}6} - \theta} \right)}}}}{A}$$T_{\beta} = {\frac{{- B} + \sqrt{B^{2} + {{AC}\mspace{14mu}{\sin\left( {{\pi\text{/}6} - \theta} \right)}}}}{A} \cdot \frac{\sin\left( {{\pi\text{/}6} + \theta} \right)}{\sin\left( {{\pi\text{/}6} - \theta} \right)}}$T₀ = T_(s) − T_(α) − T_(β) where$A = {\left( {{4u_{1\alpha}\text{/}k} - u_{o}} \right) + {\left( {{4u_{1\beta}\text{/}k} - u_{o}} \right)\frac{\sin\left( {{\pi\text{/}6} + \theta} \right)}{\sin\left( {{\pi\text{/}6} - \theta} \right)}}}$B = 4L_(o)I_(L 0) − 3u_(o)T_(s)/2 C = 8kL_(o)I_(ref)T_(s) u_(1α) is aline-to-line voltage depending on the active switching state {rightarrow over (I)}_(a), u_(1β) is a line-to-line voltage depending on theactive switching state Î_(β), k is a transformer turns ratio, u_(o) isan output voltage of the converter, θ is the angle between the referencecurrent

_(ref) and the vector with θ=0, L_(o) is an inductance of the outputinductor, I_(L0) is the current through inductor L_(o) at a beginning ofthe sampling period T_(s), T_(s) is the sampling period, and I_(ref) isa magnitude of the vector {right arrow over (I)}_(ref).
 17. A method ofclaim 14, wherein: the controller calculates the dwell times using:$T_{\alpha} = {2\sqrt{\frac{{kI}_{ref}L_{o}T_{s}{\sin\left( {{\pi\text{/}6} - \theta} \right)}}{{u_{1\;\alpha}\text{/}k} - u_{o}}}}$$T_{\beta} = {2\sqrt{\frac{{kL}_{o}I_{ref}T_{s}{\sin\left( {{\pi\text{/}6} + \theta} \right)}}{{u_{1\;\beta}\text{/}k} - u_{o}}}}$where k is a transformer turns ratio, L_(o) is an inductance of theoutput inductor, I_(ref) is the magnitude of the vector {right arrowover (I)}_(ref), T_(s) is the sampling period, θ is an angle between thereference current

_(ref) and the vector with θ=0, u_(1α) is a line-to-line voltagedepending on the active switching state {right arrow over (I)}_(α),u_(1β) is a line-to-line voltage depending on the active switching state{right arrow over (I)}_(β), and u_(o) is an output voltage of theconverter.
 18. A method of claim 11, wherein the converter is one of amatrix rectifier, a current-source rectifier, and a current-sourceinverter.
 19. A method of claim 11, further comprising operating theconverter in a continuous-conduction mode.
 20. A method of claim 11,further comprising operating the converter in a discontinuous-conductionmode.